452 F. W. Boggs and N. Tokita 
The question of the lower limit of turbulent flow is not considered by the Schlichting 
theory nor do we consider it here. Certainly, before the picture of flexible walls can be 
complete, a consideration of their influence on the fully turbulent boundary layer should be 
made. 
The theory of boundary-layer stability starts by solving the Navier-Stokes equations 
and the equation of continuity for a steady state. An arbitrary perturbation developable in a 
Fourier series is added to this solution, and the conditions are studied under which this 
arbitrary perturbation will either increase or decrease as time proceeds. If no possible per- 
turbation can increase, then the flow is assumed stable. Instability will exist if any per- 
turbation exists capable of increasing in magnitude without external excitation. Between 
these stable and unstable conditions there exists a boundary at which a perturbation neither 
increases nor decreases in magnitude. This is known as neutral stability, and it separates 
the stable from the unstable conditions. The values of the parameters of the system which 
lie on the boundary separating the regions of stable and unstable flow are known as condi- 
tions of neutral stability. The plotting of these conditions makes it possible to separate 
the regions of stable and unstable flow, and on this basis to predict when instability will 
occur. The principal aim of this paper is to express the changes in the curves of neutral 
stability which are brought about by the presence of a flexible wall, the properties of this 
wall being expressed in terms of its compliances. These compliances may not only be cal- 
culated from the nature of the walls, but they may also be measured. 
2. PERTURBATION OF TWO DIMENSIONAL PARALLEL FLOW 
If we have a flow exclusively in the x direction which, however, depends on y, then we 
may examine the effect of a small perturbation on this flow. Suppose that V, satisfies the 
Navier-Stokes equation and the equation of continuity and that it vanishes along rectilinear 
boundaries extending in the x direction. It has been shown that if the perturbation has the 
form 
+0 = 
0O(0,Y) icax-ft) ,- 
Vv =. | at e eee da (OI) 
-® 
s 
i] 
+00 
-| ia@(d,y) ef °F) ag (2.2) 
then the function ®(0,y) satisfies the Orr-Summerfeld equation 
v 2 d?v . 4 2 
(en -£)(<8- a’s) A et Syipiecnes 2H (ee - 207 Fe, a‘s| (2.3) 
a/\ dy? dy? a \dy* dy 
It is usual to express the solution of Eq. (2.3) in terms of dimensionless variables appro- 
priate to the system. Let us choose a characteristic length which may be the distance be- 
tween the edges in Couette and plane Poisseuille flows or the thickness of a boundary layer 
in a boundary-layer problem. Let us express the velocity of flow in terms of the maximum 
